Properties

Label 2.2.85.1-68.1-b3
Base field \(\Q(\sqrt{85}) \)
Conductor \((34,2 a + 16)\)
Conductor norm \( 68 \)
CM no
Base change yes: 14450.a3,34.a3
Q-curve yes
Torsion order \( 6 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{85}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 21 \); class number \(2\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 21)
 
gp: K = nfinit(a^2 - a - 21);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21, -1, 1]);
 

Weierstrass equation

\(y^2+xy=x^{3}-43x+105\)
sage: E = EllipticCurve(K, [1, 0, 0, -43, 105])
 
gp: E = ellinit([1, 0, 0, -43, 105],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -43, 105]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((34,2 a + 16)\) = \( \left(2\right) \cdot \left(17, a + 8\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 68 \) = \( 4 \cdot 17 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((2312)\) = \( \left(2\right)^{3} \cdot \left(17, a + 8\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5345344 \) = \( 4^{3} \cdot 17^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{8805624625}{2312} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(\frac{34}{9} a + \frac{58}{3} : \frac{646}{27} a + \frac{865}{9} : 1\right)$
Height \(1.12032684401892\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(8 : 13 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.12032684401892 \)
Period: \( 20.2109887435620 \)
Tamagawa product: \( 6 \)  =  \(2\cdot3\)
Torsion order: \(6\)
Leading coefficient: \(0.818656255684652\)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(17, a + 8\right) \) \(17\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\( \left(2\right) \) \(4\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 68.1-b consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base change of elliptic curves 14450.a3, 34.a3, defined over \(\Q\), so it is also a \(\Q\)-curve.